Note that c1(X)=0c_1(X) = 0 implies in general that the canonical bundle is topologically trivial. But if XX is a compact Kähler manifold, c1(X)=0c_1(X) = 0 implies further that the canonical bundle is holomorphically trivial.

The language used in this article is implicitly analytic, rather than algebraic. Is this OK? Or should I make this explicit?

If the base field is ℂ\mathbb{C}, then one can form the analyticification of XX and obtain a compactmanifold that satisfies the first given definition.

Beware that there are slighlty different (and inequivalent) definitions in use. Notably in some contexts only the trivialization of the canonical bundle is required, but not the vanishing of the H0<•<n(X,𝒪X)H^{0 \lt \bullet \lt n}(X,\mathcal{O}_X). To be explicit on this one sometimes speaks for emphasis of “strict” CY varieties when including this condition.